Necktie paradox
The necktie paradox is a puzzle or paradox within the subjectivistic interpretation of probability theory, in which two men wager over whose necktie is cheapest, with the owner of the more expensive tie giving it to the other man. Both men reason that they stand to win more than they would lose, in such a bet. The necktie paradox is a variation (and historically, the origin) of the two-envelope paradox.
Statement of paradox
Two men are each given a necktie by their respective wives as a Christmas present. Over drinks they start arguing over who has the cheaper necktie. They agree to have a wager over it. They will consult their wives and find out the prices of the neckties. The terms of the bet are that the man with the more expensive necktie has to give it to the other as the prize.
The first man reasons as follows: winning and losing are equally likely. If I lose, then I lose the value of my necktie. But if I win, then I win more than the value of my necktie. Therefore, the wager is to my advantage. The second man can consider the wager in exactly the same way; thus, paradoxically, it seems both men have the advantage in the bet. This is obviously not possible.
Resolution
The paradox can be resolved by giving more careful consideration to what is lost in one scenario ("the value of my necktie") and what is won in the other ("more than the value of my necktie"). If one assumes for simplicity that the only possible necktie prices are $20 and $40, and that a man has equal chances of having a $20 or $40 necktie, then four outcomes (all equally likely) are possible:
| Price of 1st man's tie | Price of 2nd man's tie | 1st man's gain/loss |
|---|---|---|
| $20 | $20 | 0 |
| $20 | $40 | Gain $40 |
| $40 | $20 | Lose $40 |
| $40 | $40 | 0 |
The first man has a 50% chance of a neutral outcome, a 25% chance of gaining a necktie worth $40, and a 25% chance of losing a necktie worth $40. Turning to the losing and winning scenarios: if the man loses $40, then it is true that he has lost the value of his necktie; and if he gains $40, then it is true that he has gained more than the value of his necktie. The win and the loss are equally likely, but what we call "the value of his necktie" in the losing scenario is the same amount as what we call "more than the value of his necktie" in the winning scenario. Accordingly, neither man has the advantage in the wager.
This paradox is a rephrasing of the simplest case of the two envelopes problem, and the explanation of the resolution is essentially the same.
See also
References
- Brown, Aaron C. (1995). "Neckties, Wallets, and Money for Nothing". Journal of Recreational Mathematics. 27 (2): 116–122.
- Kraitchik, Maurice (1943). Mathematical Recreations. London: George Allen & Unwin.